Abstract:If $f$ is a mod-$3$ eigenform of weight 2 and level $\Gamma_0(\ell^2)$ for a prime $\ell$ such that $\ell \equiv -1 \pmod{3}$, and $\ell$ is a vexing prime for $f$, we show that there is no obstruction to finding a minimal lift of $f$, but that there is an obstruction to finding a nonminimal lift. The key new ingredient that we prove is a modular analog of the standard character formula for a cuspidal representation of $\mathrm{GL}_2(\mathbb{F}_\ell)$, an enhancement that allows us to easily compute the group cohomology of a $3$-adic lattice in such a representation. In fact, we provide a general framework for proving such modular analogs for a broader class of representations using results of Broué and Puig in modular representation theory. We show that this class includes certain Deligne--Lusztig representations and representations coming from higher-depth supercuspidal representations of $\mathrm{GL}_2$.
From: Preston Wake [view email]
[v1]
Thu, 25 Sep 2025 11:21:36 UTC (19 KB)
[v2]
Thu, 28 May 2026 13:56:59 UTC (35 KB)
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